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VARIATION OF HILBERT COEFFICIENTS
LAURA GHEZZI, SHIRO GOTO, JOOYOUN HONG, AND WOLMER V. VASCONCELOS
Abstract. For a Noetherian local ring (R, m), the ﬁrst two Hilbert coeﬃcients, e0 and e1, of the
I-adic ﬁltration of an m-primary ideal I are known to code for properties of R, of the blowup of
Spec(R) along V (I), and even of their normalizations. We give estimations for these coeﬃcients
when I is enlarged (in the case of e1 in the same integral closure class) for general Noetherian local
rings.
1. Introduction
Let (R, m) be a Noetherian local ring of dimension d ≥ 1, and let I be an m-primary ideal. We
will consider multiplicative, decreasing ﬁltrations of R ideals,
/ = ¦I
n
[ I
0
= R, I
n+1
= II
n
, ∀n ≫0¦,
integral over the I-adic ﬁltration, conveniently coded in the corresponding Rees algebra and its
associated graded ring
!(/) =

n≥0
I
n
, gr
A
(R) =

n≥0
I
n
/I
n+1
.
Let ! =

n≥0
I
n
be the integral closure of the Rees algebra ! = !(I) with I
n
= I
n
for all n ≥ 0,
which we assume to be ﬁnite over !.
We will consider the Hilbert coeﬃcients e
i
(I) associated to m-primary ideals I, for i = 0, 1. These
integers play important roles in the corresponding blowup algebras. Some of these issues have a
long tradition in the context of Cohen-Macaulay local rings, but others are of a recent vintage for
general Noetherian local rings. From the several problem areas, we highlight the following:
(i) The comparison between e
0
and e
1
;
(ii) e
1
and normalization;
(iii) The structure of R associated to the values of e
1
;
(iv) Variation of e
i
, that is how e
i
(I) changes when I is enlarged.
We are concerned here with the last item but give brief comments on the others ﬁrst.
AMS 2010 Mathematics Subject Classiﬁcation. Primary 13A30; Secondary 13B22, 13H10, 13H15.
The ﬁrst author is partially supported by a grant from the City University of New York PSC-CUNY Research
Award Program-41. The second author is partially supported by Grant-in-Aid for Scientiﬁc Researches (C) in Japan
(19540054) and by a grant from MIMS (Meiji Institute for Advanced Study of Mathematical Sciences). The last
author is partially supported by the NSF.
1
2 LAURA GHEZZI, SHIRO GOTO, JOOYOUN HONG, AND WOLMER V. VASCONCELOS
(i) For Cohen-Macaulay rings, an uniform bound for e
1
(I) ﬁrst appeared for rings of dimension 1
in the work of D. Kirby ([K]),
e
1
(m) ≤
_
e
0
(m)
2
_
.
Progressively, quadratic bounds of this type were developed for arbitrary m-primary ideals in all
dimensions by several authors. As a basic source, [RV2] has a systematic development of these
formulas along with a comprehensive bibliography. It also treats more general ﬁltrations which
were helpful to us. Among the formulas which more directly inﬂuenced the authors here, we single
out two developed in the work of J. Elias ([E1, E2]) and M. E. Rossi and G. Valla ([RV1, RV2]).
For an d-dimensional Cohen-Macaulay local ring and an m-primary ideal I, the ﬁrst bound asserts
that if I is minimally generated by m = ν(I) elements,
e
1
(I) ≤
_
e
0
(I)
2
_
−
_
m−d
2
_
−λ(R/I) + 1.
The other bound uses the m-adic order of I, that if I ⊂ m
s
and I ,= m
s
, then
e
1
(I) ≤
_
e
0
(I) −s
2
_
.
Recently, K. Hanumanthu and C. Huneke ([HH]) brought a new parameter to bear on these formulas
with their proof that
e
1
(I) ≤
_
e
0
(I) −k
2
_
,
where k is the maximal length of chains of integrally closed ideals between I and m.
(ii) Since e
1
(I) ≤ e
1
(I) := e
1
(!), bounds with a diﬀerent character arise. A baseline is the fact
that when R is analytically unramiﬁed, but not necessarilly Cohen-Macaulay, one has e
1
(I) ≥ 0
([GHM]). An upper bound for e
1
(I) (see [PUV] for other bounds) is the following. Let (R, m) be
a reduced Cohen-Macaulay local ring of dimension d, essentially of ﬁnite type over a perfect ﬁeld,
and let I be an m-primary ideal. Let δ be a regular element of the Jacobian ideal of R. Then
e
1
(I) ≤ e
1
(I) ≤
t
t + 1
_
(d −1)e
0
(I) + e
0
((I + δR)/δR)
¸
,
where t is the Cohen-Macaulay type of R. In particular, if R is a regular local ring
e
1
(I) ≤
(d −1)e
0
(I)
2
.
(iii) When R is not Cohen-Macaulay, the issues become less structured since the values of e
1
(I)
may be negative. In fact, using the values of e
1
(I) for ideals generated by systems of parameters
led to the characterization of several properties (Cohen-Macaulay, Buchsbaum, ﬁnite cohomology)
of the ring R itself (see [GhGHOPV], [GhHV], [GO], [MV], [V2]).
(iv) We shall now outline the main results of this note.(We refer to [V1] for basic deﬁnitions and
Rees algebras theory.) Sections 2 and 3 are organized around a list of questions about the changes
that e
0
(I) and e
1
(I) undergo when I varies. An important case is
e
0
(J), e
1
(J) −→e
0
(I), e
1
(I), I = (J, x).
VARIATION OF HILBERT COEFFICIENTS 3
Clearly the optimal baseline is that of an ideal J generated by a system of parameters, but we
will consider very general cases. As will be seen, some relationships involve the multiplicity f
0
(J)
of the special ﬁber. To describe one of these estimates, let (R, m) be a Noetherian local ring of
dimension d ≥ 1, let J be an m-primary ideal and let I = (J, h
1
, . . . , h
m
) be integral over J of
reduction number s = red
J
(I). Then Theorem 2.6 asserts that
e
1
(I) −e
1
(J) ≤ λ(R/(J : I))
__
m + s
s
_
−1
_
f
0
(J),
where f
0
(J) is the multiplicity of the special ﬁber of !(J) =

r=1
ν(J
n−r
).
VARIATION OF HILBERT COEFFICIENTS 5
Now for n ≫ 0, λ(R/J
n
) − λ(R/I
n
) is the diﬀerence of two polynomials of degree d and with
same leading (binomial) coeﬃcients e
0
(J) and e
0
(I), therefore it is at most a polynomial of degree
d −1 and leading coeﬃcient e
1
(I) −e
1
(J). On the other hand, for n ≫0, we have
λ(R/(J : h))
s

the issues become less structured since the values of e1 (I) may be negative. RV2]). [GO]. an uniform bound for e1 (I) ﬁrst appeared for rings of dimension 1 in the work of D. In fact. E2]) and M.
. JOOYOUN HONG. the ﬁrst bound asserts that if I is minimally generated by m = ν(I) elements. AND WOLMER V. [GhHV]. In particular. 2 2 e0 (I) − s . (ii) Since e1 (I) ≤ e1 (I) := e1 (R).2
LAURA GHEZZI. 2
The other bound uses the m-adic order of I. As a basic source. essentially of ﬁnite type over a perfect ﬁeld. Let (R. For an d-dimensional Cohen-Macaulay local ring and an m-primary ideal I. one has e1 (I) ≥ 0 ([GHM]). Then t (d − 1)e0 (I) + e0 ((I + δR)/δR) . we single out two developed in the work of J. m) be a reduced Cohen-Macaulay local ring of dimension d. Among the formulas which more directly inﬂuenced the authors here. Let δ be a regular element of the Jacobian ideal of R. An important case is e0 (J). that if I ⊂ ms and I = ms . Elias ([E1. VASCONCELOS
(i) For Cohen-Macaulay rings. Hanumanthu and C. I = (J. It also treats more general ﬁltrations which were helpful to us. x). e1 (I). E. [V2]). quadratic bounds of this type were developed for arbitrary m-primary ideals in all dimensions by several authors. (iv) We shall now outline the main results of this note. Huneke ([HH]) brought a new parameter to bear on these formulas with their proof that e0 (I) − k e1 (I) ≤ . Buchsbaum. e1 (I) ≤ e1 (I) ≤ t+1 where t is the Cohen-Macaulay type of R. using the values of e1 (I) for ideals generated by systems of parameters led to the characterization of several properties (Cohen-Macaulay. e1 (I) ≤ e0 (I) m−d − − λ(R/I) + 1.) Sections 2 and 3 are organized around a list of questions about the changes that e0 (I) and e1 (I) undergo when I varies. [RV2] has a systematic development of these formulas along with a comprehensive bibliography. 2
(iii) When R is not Cohen-Macaulay. SHIRO GOTO. K. but not necessarilly Cohen-Macaulay. An upper bound for e1 (I) (see [PUV] for other bounds) is the following. 2 where k is the maximal length of chains of integrally closed ideals between I and m. ﬁnite cohomology) of the ring R itself (see [GhGHOPV]. then e1 (I) ≤
Recently. if R is a regular local ring e1 (I) ≤ (d − 1)e0 (I) . Rossi and G. and let I be an m-primary ideal. e0 (m) e1 (m) ≤ . 2 Progressively. A baseline is the fact that when R is analytically unramiﬁed. bounds with a diﬀerent character arise. Valla ([RV1. [MV]. Kirby ([K]). e1 (J) −→ e0 (I).(We refer to [V1] for basic deﬁnitions and Rees algebras theory.

but we will consider very general cases. If n = 1. then M ≃ R/m and the assertion is clear. h) be m–primary ideals of R.
.2. Lemma 2. . let (R. so that λ(M ⊗ N ) ≤ λ(L ⊗ N ) + λ((M/L) ⊗ N ) ≤ (1 + (n − 1))·ν(N ) = λ(M )ν(N ). m) be a Noetherian local ring of dimension d and let J ⊂ I = (J. . Suppose that n ≥ 2 and choose an R–submodule L of M with λ(L) = 1.VARIATION OF HILBERT COEFFICIENTS
3
Clearly the optimal baseline is that of an ideal J generated by a system of parameters.6 asserts that e1 (I) − e1 (J) ≤ λ(R/(J : I))· m+s − 1 ·f0 (J). Let (R. Proof. the induction hypothesis shows λ((M/L) ⊗ N ) ≤ (n − 1)ν(N ). As will be seen. Upper bounds for the variations of e0 (I) and e1 (I) In our calculations we make repeated use of the following elementary observation.1. Then e0 (J) − e0 (I) ≤ λ(R/(J : I))·f0 (J). and vanishes when R is Cohen–Macaulay. Then Theorem 2. we address the need to link the value of redJ (I) to other properties of J. 2. . m) is a Noetherian local ring and M is an R–module of ﬁnite length λ(M ). e1 (J) is always non-positive. where ν(N ) denotes the minimal number of generators for N . When J is a minimal reduction of I.3: Let (R. For an m-primary ideal I and a minimal reduction J of I. . according to [MSV]. but we give a general formulation in Theorem 3. s
where f0 (J) is the multiplicity of the special ﬁber of R(J) = n≥0 J n . In Section 3. By tensoring 0 → L → M → M/L → 0 with N . Induct on n = λ(M ). This is a well-known fact when R is a Cohen-Macaulay ring. To describe one of these estimates. 0}. we get the exact sequence L ⊗ N → M ⊗ N → (M/L) ⊗ N → 0. let J be an m-primary ideal and let I = (J. there exists a minimal reduction Q of I such that redQ (I) ≤ max{d·λ(R/J) − 2d + 1. some relationships involve the multiplicity f0 (J) of the special ﬁber. ✷ Theorem 2. Since λ(M/L) = n − 1. m) be a Noetherian local ring of dimension d ≥ 1 and inﬁnite residue ﬁeld. m) be a Noetherian local ring of dimension d ≥ 1. In fact. h1 . hm ) be integral over J of reduction number s = redJ (I). If (R. for unmixed local rings the vanishing characterizes Cohen-Macaulayness ([GhGHOPV]). We add a word of warning in reading some of the formulas with terms like e1 (I) − e1 (J). then λ(M ⊗ N ) ≤ λ(M )·ν(N ) for every ﬁnitely generated R–module N .